A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than
1 and itself. A natural number greater than 1 that is not a prime number is called a composite number.
For example, 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2
and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of
primes in number theory: any integer greater than 1 can be expressed as a product of primes that is
unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because it is the
The property of being prime is called primality. A simple but slow method of verifying the primality of
a given number n is known as trial division. It consists of testing whether n is a multiple of any integer
between 2 and . Algorithms that are much more efficient than trial division have been devised to test the
primality of large numbers. Particularly fast methods are available for primes of special forms, such as
Mersenne primes. As of 2011, the largest known prime number has nearly 13 million decimal digits.
There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known useful
formula that yields all of the prime numbers and no composites. However, the distribution of primes,
that is to say, the statistical behaviour of primes in the large, can be modeled.
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The first result in that direction is the prime number theorem, proven at the end of the 19th century,
which says that the probability that a given, randomly chosen number n is prime is inversely
proportional to its number of digits, or the logarithm of n. Many questions around prime numbers
remain open, such as Goldbach's conjecture, which asserts that every even integer greater than 2 can be
expressed as the sum of two primes, and the twin prime conjecture, which says that there are infinitely
many pairs of primes whose difference is 2. Such questions spurred the development of various
branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in
several routines in information technology, such as public-key cryptography, which makes use of
properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give
rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements
and prime ideals.
Definition and examples ;- A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime or a prime
number if it is greater than 1 and has exactly two divisors, 1 and the number itself. Natural numbers
greater than 1 that are not prime are called composite. The number 12 is not a prime, as 12 items can be
placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into
several equal-size columns of more than 1 item each, there will always be some extra items left (a
remainder). Therefore the number 11 is a prime. Among the numbers 1 to 6, the numbers 2, 3, and 5 are
the prime numbers, while 1, 4, and 6 are not prime. 1 is excluded as a prime number, for reasons
explained below. 2 is a prime number, since the only natural numbers dividing it are 1 and 2. Next, 3 is
prime, too: 1 and 3 do divide 3 without remainder, but 3 divided by 2 gives remainder 1. Thus, 3 is
prime. However, 4 is composite, since 2 is another number (in addition to 1 and 4) dividing 4 without
4 = 2 · 2.
5 is again prime: none of the numbers 2, 3, or 4 divide 5. Next, 6 is divisible by 2 or 3, since
6 = 2 · 3. Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. In
general, no even number greater than 2 is prime: any such number n has at least three distinct divisors,
namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any
prime number greater than 2. In a similar vein, all prime numbers bigger than 5, written in the usual
decimal system, end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or
5 are multiples of 5.
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